Navier-Stokes Equations

\[ \frac{\partial}{\partial t}[\rho\mathbf{v}] + \nabla \cdot \{\rho\mathbf{v}\mathbf{v}\} = -\nabla p + \nabla \cdot \left\{\mu\left[\nabla\mathbf{v} + (\nabla\mathbf{v})^{\text{T}}\right]\right\} \]

\[ \frac{\partial\rho}{\partial t} +\nabla\cdot (\rho\mathbf{u}_{}) = 0 \]

Species Transport &

Gauss law

\[ \frac{ \partial c_{i} }{ \partial t } = \nabla \cdot(D_{i}\nabla c_{i}) + \nabla \cdot [( \underbrace{ D_{i} \dfrac{ez_{i}}{kT} } _{ \text{Stokes-Einstein relation} }\nabla \Psi)c_{i} ] \]

\[ \nabla^2 \Psi _{E} =\frac{-\rho_E}{\epsilon\epsilon_{0}} = \frac{-F \sum_{i} z_{i} c_{i}}{ \epsilon\epsilon_{0}} \]